The Tora tells us (Dvarim/Deuteronomy 11,13):
… לְאַהֲבָה אֶת-יְהוָה אֱלֹהֵיכֶם, וּלְעָבְדוֹ, בְּכָל-לְבַבְכֶם … … to love the LORD your God, and to serve Him with all your heart …
The Talmud tells us (Taanit 2b, Soncino translation, sorry for the incoherence)
Rain is to be made in the Prayer? – It has been taught: To love the Lord your God and to serve Him with all your heart. What is Service of Heart? You must needs say, Prayer. And the verse following reads, That I will give the rain of your land in its season, the former rain and the latter rain.
All the specifics, how often each day (if each day?), what words to say, etc. are rabbinic. This is the usual source offered, although there are alternatives (Rabbi Shais Taub at Askmoses: Is it a Mitzvah (commandment) to pray?).